Abstract and Applied Analysis

Volume 2013 (2013), Article ID 290287, 2 pages

http://dx.doi.org/10.1155/2013/290287

Letter to the Editor

Periodic Solution of the Hematopoiesis Equation

Ji-Huan He

National Engineering Laboratory for Modern Silk, College of Textile and Clothing Engineering, Soochow University, 199 Ren-Ai Road, Suzhou 215123, China

Received 4 December 2012; Accepted 30 December 2012

Copyright © 2013 Ji-Huan He. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Wu and Liu (2012) presented some results for the existence and uniqueness of the periodic solutions for the hematopoiesis model. This paper gives a simple approach to find an approximate period of the model.

Wu and Liu studied the following hematopoiesis model [1]:

where denotes the density of mature cells in blood circulation. The physical meaning of other parameters is referred to [1].

Equation (1) admits periodic solutions as revealed in [1]. Hereby we suggest a simple approach to the search for an approximate period of (1) using a simple amplitude-frequency formulation [2–5]. To this end, we rewrite (1) in the form

Assume that the periodic solution can be expressed in the form

Submitting (3) into (2) results in the following residual:

In order to use the amplitude-frequency formulation [2–5], we choose two trial frequencies and locate them at .

Setting ,  , and , , respectively, we have

The frequency can be then obtained approximately in the form [2–5]

This formulation has been widely used to solve periodic solutions of various nonlinear oscillators [6–13], and it is often called as He’s frequency formulation, He’s amplitude-frequency formulation, or He’s frequency-amplitude formulation. In case , no period solution is admitted. A similar criterion is given for a nonlinear equation arising in electrospinning process [14].

Acknowledgments

The work is supported by PAPD (a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions), National Natural Science Foundation of China under Grant no. 10972053, and Project for Six Kinds of Top Talents in Jiangsu Province, China (Grant no. ZBZZ-035).

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